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Properties

Label 2-21-21.11-c4-0-2
Degree $2$
Conductor $21$
Sign $-0.221 - 0.975i$
Analytic cond. $2.17076$
Root an. cond. $1.47335$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 7.79i)3-s + (−8 + 13.8i)4-s + (35.5 + 33.7i)7-s + (−40.5 − 70.1i)9-s + (−72 − 124. i)12-s + 191·13-s + (−127. − 221. i)16-s + (300.5 + 520. i)19-s + (−423 + 124. i)21-s + (−312.5 + 541. i)25-s + 729·27-s + (−752 + 221. i)28-s + (876.5 − 1.51e3i)31-s + 1.29e3·36-s + (−1.29e3 − 2.24e3i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.724 + 0.689i)7-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)12-s + 1.13·13-s + (−0.499 − 0.866i)16-s + (0.832 + 1.44i)19-s + (−0.959 + 0.282i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (−0.959 + 0.282i)28-s + (0.912 − 1.57i)31-s + 36-s + (−0.946 − 1.63i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.221 - 0.975i$
Analytic conductor: \(2.17076\)
Root analytic conductor: \(1.47335\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :2),\ -0.221 - 0.975i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.626337 + 0.784533i\)
\(L(\frac12)\) \(\approx\) \(0.626337 + 0.784533i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 + (-35.5 - 33.7i)T \)
good2 \( 1 + (8 - 13.8i)T^{2} \)
5 \( 1 + (312.5 - 541. i)T^{2} \)
11 \( 1 + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 - 191T + 2.85e4T^{2} \)
17 \( 1 + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-300.5 - 520. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 7.07e5T^{2} \)
31 \( 1 + (-876.5 + 1.51e3i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (1.29e3 + 2.24e3i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 2.82e6T^{2} \)
43 \( 1 - 23T + 3.41e6T^{2} \)
47 \( 1 + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-983 - 1.70e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-4.40e3 + 7.62e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 2.54e7T^{2} \)
73 \( 1 + (-624.5 + 1.08e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-6.18e3 - 1.07e4i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 4.74e7T^{2} \)
89 \( 1 + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 1.88e4T + 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.66060956208785914369390168311, −16.54033783760019777619329157182, −15.43041620240432757634491334678, −13.95995845709668567419156554226, −12.26054312942087646333414344231, −11.21909595757549120959511927269, −9.428429490297320892806909316934, −8.134032882213174129328589700882, −5.58141662467568535731688105827, −3.82602547844482866508531298386, 1.09447029849217569637223041483, 4.99130051023010146226522750460, 6.66159774954992762152960013723, 8.450007410165770593327849381142, 10.45298405814438869166039595380, 11.56977155817409141777589373308, 13.43413179410918312099670764471, 14.05314343381431385109786083765, 15.78206950005915816353246692068, 17.50039957039732988858801580810

Graph of the $Z$-function along the critical line